## Abstract

For any n-by-n matrix A, we consider the maximum number k=k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary ∂W(A) of the numerical range W(A) of A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B⊕C, we show that k(A)=2 if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C and k(C)=2. For an irreducible matrix A, we can determine exactly when the value of k(A) equals the size of A. These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W(A).

Original language | English |
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Pages (from-to) | 2584-2597 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 9 |

DOIs | |

State | Published - 1 Nov 2013 |

## Keywords

- Compression
- Direct sum
- Numerical range